There's a little bit of information in a tangentially-related thread
here. There are lots of things which you can't affect- British criticism of American racism, the virulent anti-British tone of the Northern press, the Morrill tariff- but there are a couple of things which Lincoln could have done.
The first is not making Seward Secretary of State. Appointing as the man in charge of foreign relations an individual who frequently threatens to invade Canada is not an act designed to get Britain on side. Furthermore, the US would have come out of the Trent Affair a lot better in British eyes if they'd admitted that Wilkes acted incorrectly rather than trying to score political points by conflating the issue with impressment as Seward did. Whether there's an alternative post for him is a question for people who are more au fait with the internal politics of the Republican party than I am.
The second is the issue of slavery, and here you tread a delicate line because any act which gets Britain on side by its very nature alienates the border states. Had Lincoln not jettisoned Fremont (excuse the lack of an acute accent), the British would have been more inclined to put faith in his anti-slavery credentials. Another step would have been immediately to offer Britain the right of search over American ships for the duration of the conflict, on the grounds that the US Navy squadron stationed off the coast of Africa would be better served at home. Britain in general, and Palmerston in particular, have been campaigning for the right for literally decades. The extent to which the granting of the right in April 1862 changes his opinion can be demonstrated by the following two quotes:
"I regret to say, as my noble Friend has stated, that we have not received from the Government of the United States that assistance which we were entitled to expect from a Government of free men. The Government of the United States have taken engagements as well as the Spanish nation. They are bound by treaty to co-operate with England for the suppression of the slave trade by stationing a certain amount of naval force upon the coast of Africa. That engagement has been more or less fulfilled from time to time, but the American Government have prevented British cruisers from meddling with ships sailing under the American flag, except at their risk and peril, in the event of the ships being found not only to have the American flag, but to have American papers proving their nationality." 26th February 1861
"VISCOUNT PALMERSTON said, he rose to move for leave to introduce a Bill to carry into effect the treaty between Her Majesty and the United States of America for the suppression of the African slave trade. The American Government had behaved in the handsomest manner in reference to the treaty. They had long been aware that the American flag had been perverted as a cover for carrying on the slave trade, and sensible of the evil and inconvenience to which that practice had given rise, they had of their own accord proposed the treaty, which was in all respects adapted, as far as their flag was concerned, to put an end to the perpetration of the crime." 19th June 1862